Rosenhouse on Amanda Shaw

Following up on his previous post, “Is Math a Gift From God?” — calculus students say, “No!” — Jason Rosenhouse has a new essay for your delectation, “Is God Like an Imaginary Number?” Again, the short answer is, “Nope.” The longer answer will take us into the history of mathematics, the role of mysticism in theology and the relationship between science and verbal description.

Rosenhouse sets himself the task of fisking an essay in the religious periodical First Things, by a “Junior Fellow” of that publication named Amanda Shaw. Shaw’s thesis is that the notion of God is akin to that of an imaginary number, and moreover that the same closed-minded orthodoxy which rejected the latter from mathematics for oh so many years is unjustly keeping the former out of science. I find this stance to be, in a word, ironical: if you’re looking for dogmatism and condemnations of the heterodox, your search will be much more rewarding if you look among the people who reject scientific discoveries because they are inconsistent with a Bronze Age folk tale than if you search through science itself!

Still, it’s a fun chance to talk about history and mathematics.

PART A: COMPLEX NUMBERS

As I described earlier, “imaginary” and “complex” numbers arise naturally when you think about the ordinary, humdrum “real numbers” — you know, fractions, decimals and all those guys — as lengths on a number line. In this picture, adding two numbers corresponds to sticking line segments end-to-end, multiplication means stretching or squishing (in general, scaling) line segments, and negation means flipping a segment over to lie on the opposite side of zero. Complex numbers appear when you ask the question, “What operation, when performed twice in succession upon a line segment, is equivalent to a negation?” Answer: rotating by a quarter-turn!

Historically, mathematicians started getting into complex numbers when they tried to find better and better ways to solve real-number equations. Girolamo Cardano (1501–1576), also known as Jerome Cardan, posed the following problem:

If some one says to you, divide 10 into two parts, one of which multiplied into the other shall produce […] 40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion.

Writing this in more modern algebraic notation, this is like saying [tex] x + y = 10 [/tex] and [tex] xy = 40 [/tex], which we can combine into one equation by solving for [tex] y [/tex], thusly:

[tex] xy = x(10 – x) = 40.[/tex]

In turn, shuffling the symbols around gives

[tex] x^2 – 10x + 40 = 0,[/tex]

which plugging into ye old quadratic formula yields

[tex] x = \frac{10 \pm \sqrt{100 – 160}}{2}, [/tex]

or, boiling it down,

[tex] x = 5 \pm \sqrt{-15}. [/tex]

Totally loony! Taking the square root of a negative number? Forsooth, thy brains are bubbled! Oh, wait, didn’t we just realize that we could maybe handle the square root of a negative number by moving into a two-dimensional plane of numbers? Yes, we did: that’s the prize our talk of flips and rotations won us!
Cardano realized that you could manipulate numbers of the form [tex]5 \pm \sqrt{-15}[/tex], multiplying them and adding them together in reasonable ways, such that when you got a solution of such a form, you could plug it back into the original equations and find that your work “checked.” (“Substitute your answer into the original equation to check your work.” Remember being told to do that on your algebra homework? Man, I sure do.) Cardano’s verdict was that this was “so subtle that it is useless.”

How wrong can you get?

Complex numbers now find applications all over the place. Modern physics honestly couldn’t be done without them: they are a fantastic labor-saving device for studying vibrations and oscillations in classical mechanics, and they appear to be built into the fundamental structure of quantum mechanics. Electrical engineers use them to figure out what circuits will do when you send alternating current through them. Square roots of negative numbers: take them, for fun and profit!

PART B: INTERLUDE ON ASTROLOGY

I’ll take a moment to note that Shaw says, “Calling Cardano a scientist is both apt and preposterous.” Why “preposterous”? Well, because he studied astrology. Today, we know beyond a doubt that astrology is bunk. People who “work” on it are not doing science — nor, incidentally, are they advancing the moral state of our civilization, but that’s a different matter. Surely, Cardano is tainted by the mere association with this backward pseudoscience?

Ah, but look at his dates! Born in 1501, died in 1576 — a full generation before Kepler even began the work which would turn astronomy into a true science, tearing it forever away from astrological nonsense. Can we really judge Cardano without judging his time as well? And however reprehensible astrology might have been in the sixteenth century, it is infinitely more so now, and thus any rebuke which we might cast upon Cardano’s generation must apply far more forcefully to those who peddle antiscience today.

PART C: PROGRESS IN SCIENCE AND THEOLOGY

When you have even a small familiarity with the practical uses of mathematical concepts, and even a glancing acquaintance with the beauty of abstract interrelationships, witnessing the abuse of mathematics to give a false respectability to mystical mumblings becomes a truly painful experience. Jason Rosenhouse writes,

Complex numbers (of which the pure imaginary numbers are a subset) earned their acceptance first by proving their usefulness over and over again, and second by being placed on a firm logical footing by various mathematicians. Nowadays complex numbers are not one wit more mysterious than real numbers or rational numbers or even just ye olde counting numbers.

It was logic, reason and hard work that showed the need for complex numbers, and more logic, more reason and more hard work that transformed them from gibberish into a useful tool.

Rosenhouse goes on to say, criticizing the thesis of his fiskee,

Compare that to God. Back in the days when everything in nature was mysterious and unpredictable, it made sense to invoke unfathomable gods to explain it all. But the march of scientific progress, far from showing the need for invoking God as an explanation, has actually gone in the exact opposite direction.

My only quibble with this statement is that, back in the day, gods were fathomable, to an extent: they were, after all, bigger and more powerful versions of ourselves. The edge of theology has advanced into declarations of unfathomability, starting with the Trinity endorsed at Nicaea, but even today, the billboards and red-letter bibles spread the message that the ultimate order of everything hinges upon human emotions and even familial relationships — for God so loved the world — ideas which are not too many fathoms deep.

Indeed, the game of theology has been to construct a god whose presence is less and less distinguishable from that of no god at all. By contrast, science and mathematics have spent those same generations developing ideas of greater and greater refinement and applying them to observations of the physical world, observations of continually increasing sophistication. As it is the latter relationship which reflects the complexity of Nature, could we but expect that the ideas of the latter would evince the greater subtlety?

Ahem.

PART D: WORD GAMES

Rosenhouse says, “Shaw is just playing word games here.” I’ve noticed that “word games” seem to be a common and perhaps underappreciated form of sophistry, where pseudoscience and pseudomathematics are concerned. You can’t reason about scientific and mathematical topics using the everyday meanings of words and verbal or “literary” modes of thinking; equations are not merely decorations adorning a prose narrative. Alan Sokal and Jean Bricmont give the example of a social-science friend who came to them one day and asked, very puzzled, how it was that quantum mechanics could say the world was both discontinous and interconnected. A few days ago, we had a troll at Pharyngula who, insisting that only “Theism” could solve the problem of dark matter, displayed some rather confused ideas about “energy” and “gravity.” Using the actual physics, energy is interchangeable with mass — that’s the content of Einstein’s famous equation [tex] E = mc^2 [/tex] — and we’ve known since Newton that mass is the source of gravitational attraction. Of course, we’ve updated the Newtonian force law

[tex] \vec{F} = -\frac{GMm}{r^2} \hat{r} [/tex]

to the Einsteinian description of curved spacetime, but still, more energy means more mass which means a stronger gravitational pull. A sealed box of hot helium gas will actually pull very slightly harder on nearby objects than a box of the same size containing the same number of helium atoms at a lower temperature. Yet to our persistent Pharyngula commenter, “energy” and “gravity” were diametrically opposed concepts — because, I think, when you have “lots of energy” you’re not “weighed down” but instead “bouncing around the room.”

Inconsistencies which appear at the verbal level turn out to be illusory when you progress to the actual science.